Stochastic frozen-flow methods for the intrinsic integration of stochastic dynamics on Riemannian manifolds (Adrien Busnot Laurent, INRIA Rennes)

11.02.2025 14:00

In stochastic optimization, molecular dynamics, quantum physics, or in the training of neural networks, one is often interested in sampling from the law of a stochastic process. These dynamics are often subject to geometric constraints (fixed distance between particles, Physics Informed Neural Networks, ...). In this context, it is crucial to develop numerical approaches that take into account both the geometric and random features of the dynamics. The literature uses penalty formulations or extrinsic numerical integrators. These solutions are costly, subject to significant time-step restrictions, and require formulation of the dynamics in much higher-dimensional spaces.
In this talk, we present a new class of methods that rely on intrinsic geometric operations (in the spirit of Crouch-Grossman methods), a new robust convergence analysis, and an algebraic formalism of planar exotic Butcher series for the foundation of the weak order theory of intrinsic stochastic methods. A second order integrator is presented and tested numerically for a handful of manifolds.
This is joint work with Eugen Bronasco and Baptiste Huguet.

Lieu

Conseil Général 7-9, Room 1-05, Séminaire d'analyse numérique

entrée libre